When changing a dollar bill, you can give 1 coin (1 silver dollar), 2
coins (2 half-dollars), 3 coins (2 quarters and 1 half-dollar), and so
on. What is the least positive number of coins that is impossible to give
as change for a dollar bill?

If circles packed in a 100 by 100 square are repacked so that the centers
of any three tangent circles form an equilateral triangle, what is the
maximum number of additional circles that can be packed?

Doctor Greenie answers a chestnut about repeated division and
remainders, first working the question forwards before using the
inverse of a function to solve the same problem backwards much more
easily.

Each of four rows of coins has exactly one penny, one nickel, one
dime, and one quarter. No row, either horizontal, vertical, or
diagonal, has more than one coin of each kind. How are the coins
arranged?

Use mathematical induction to prove that for any positive integer n,
if any one square is removed from a 2^n x 2^n checkerboard, then the
remaining squares can be completely (and exactly) covered with
L-shaped pieces composed of three squares.

A challenging logic problem involving five criminals charged with five
crimes. The names of the criminals are the same as the crimes, but no
criminal commited the crime of his name. Using several clues,
determine who committed murder.

You want to send a valuable object to a friend. You have a box and
several locks with keys. But your friend does not have the key to any
lock that you have, and any key you send might be copied. How can you
send to object safely?

A student sees a palindrome in the date 01 02 2010, and wonders how to
generate all such palindromic dates. Building on another math doctor's
work with date arithmetic, Doctor Carter shares a program written in C,
then goes on to explain the purpose of each line of code.